Long-Run Risk

Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

A common belief is that the law of averages will rule in the long run, eliminating risk.

It’s true that the average outcome from a gamble should not be risky in the long run.

But the gain or loss from gambling is average gain per gamble $\times$ number of gambles.

Betting on the stock market

Based on history, the bet is definitely in our favor. We have much better odds than in a casino.
If we play for a long time, we will almost certainly come out ahead.
But how far ahead is quite uncertain.
- In the worst 20-year period in the U.S. stock market since 1926, $1 $\rightarrow$ $1.73, a geometric average return of 2.8% per year (1929-1948).
- In the best 20-year period since 1926, $1 $\rightarrow$ $24.65, a geometric average return of 17.4% per year (1980-1999).

import numpy as np

mn = 0.125
sd = 0.174
nyears = 20

r = np.random.normal(loc=mn, scale=sd, size=nyears)
comp_ret = np.prod(1+r)

nsims = 1000

r = np.random.normal(loc=mn, scale=sd, size=nyears*nsims)
r = r.reshape((nyears, nsims))
comp_ret = np.prod(1+r, axis=0)

mean     34.52
std      35.50
min       1.36
10%       7.43
25%      13.07
50%      23.06
75%      43.25
90%      72.34
max     317.80
dtype: float64

Uncertainty about long-run returns $\Rightarrow$ uncertainty about retirement plans.

We can revisit the retirement plan, but

generate random returns and simulate many lifetimes.